Subgroups — Q₈×#6891 #14517836

Order 1944.

756 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6891	C₂	1	3	yes	maximal
2	C₃⁴×Q₈	C₃	1	1	yes	maximal, characteristic
3	C₃×Q₈×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
4	Q₈×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₂×#6891	C₂²	1	1	yes	characteristic
6	C₃⁴×C₄	C₂×C₃	1	3	yes	—
7	C₃×C₄×3₋¹⁺²	C₂×C₃	1	9	yes	—
8	C₄×C₃²⋊C₉	C₂×C₃	1	27	yes	—
9	#6891	Q₈	1	1	yes	characteristic, direct product factor, Sylow
10	C₃³×Q₈	—	3	3	no	—
11	C₃³×Q₈	—	3	9	no	—
12	C₃³×Q₈	C₃²	1	1	yes	characteristic
13	C₃³×Q₈	C₃²	1	3	yes	—
14	Q₈×3₋¹⁺²	—	3	9	no	—
15	C₃×Q₈×C₉	C₃²	1	9	yes	complemented
16	C₂×C₃⁴	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	3	yes	—
18	C₂×C₃²⋊C₉	C₂²×C₃	1	9	yes	—
19	C₃³×C₄	—	3	9	no	—
20	C₃³×C₄	—	3	27	no	—
21	C₃³×C₄	C₂×C₃²	1	3	yes	—
22	C₃³×C₄	C₂×C₃²	1	9	yes	—
23	C₄×3₋¹⁺²	—	3	27	no	—
24	C₃×C₄×C₉	C₂×C₃²	1	27	yes	—
25	C₃⁴	C₃×Q₈	1	1	yes	characteristic
26	C₃×3₋¹⁺²	C₃×Q₈	1	3	yes	complemented
27	C₃²⋊C₉	C₃×Q₈	1	9	yes	complemented
28	C₃²×Q₈	—	3	27	no	—
29	C₃²×Q₈	—	3	3	no	—
30	C₃²×Q₈	—	3	3	no	—
31	C₃²×Q₈	—	3	6	no	—
32	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
33	C₃²×Q₈	3₋¹⁺²	1	9	yes	complemented
34	C₃²×Q₈	C₃³	1	1	yes	characteristic
35	Q₈×C₉	—	3	9	no	—
36	C₂×C₃³	—	3	3	no	—
37	C₂×C₃³	—	3	9	no	—
38	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
39	C₂×C₃³	C₂²×C₃²	1	3	yes	—
40	C₂×3₋¹⁺²	—	3	9	no	—
41	C₂×C₃×C₉	C₂²×C₃²	1	9	yes	—
42	C₃²×C₄	—	3	81	no	—
43	C₃²×C₄	—	3	9	no	—
44	C₃²×C₄	—	3	9	no	—
45	C₃²×C₄	—	3	18	no	—
46	C₃²×C₄	C₂×H(3,𝔽₃)	1	9	yes	—
47	C₃²×C₄	C₂×3₋¹⁺²	1	27	yes	—
48	C₃²×C₄	C₂×C₃³	1	3	yes	—
49	C₄×C₉	—	3	27	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	3	9	no	—
52	C₃³	C₃²×Q₈	1	1	yes	characteristic
53	C₃³	C₃²×Q₈	1	3	yes	—
54	3₋¹⁺²	—	3	9	no	—
55	C₃×C₉	C₃²×Q₈	1	9	yes	complemented
56	C₃×Q₈	—	3	3	no	—
57	C₃×Q₈	—	3	9	no	—
58	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃×Q₈	C₃×3₋¹⁺²	1	3	yes	—
60	C₂×C₃²	—	3	27	no	—
61	C₂×C₃²	—	3	3	no	—
62	C₂×C₃²	—	3	3	no	—
63	C₂×C₃²	—	3	6	no	—
64	C₂×C₃²	C₂²×H(3,𝔽₃)	1	3	yes	—
65	C₂×C₃²	C₂²×3₋¹⁺²	1	9	yes	—
66	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
67	C₂×C₉	—	3	9	no	—
68	C₃×C₄	—	3	9	no	—
69	C₃×C₄	—	3	27	no	—
70	C₃×C₄	C₂×C₃×H(3,𝔽₃)	1	3	yes	—
71	C₃×C₄	C₂×C₃×3₋¹⁺²	1	9	yes	—
72	C₃²	—	3	27	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	6	no	—
76	C₃²	Q₈×H(3,𝔽₃)	1	3	yes	—
77	C₃²	Q₈×3₋¹⁺²	1	9	yes	complemented
78	C₃²	C₃³×Q₈	1	1	yes	characteristic
79	C₉	—	3	9	no	—
80	Q₈	#6891	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×C₃	—	3	3	no	—
82	C₂×C₃	—	3	9	no	—
83	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	1	yes	characteristic
84	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	3	yes	—
85	C₄	C₂×#6891	1	3	yes	—
86	C₃	—	3	3	no	—
87	C₃	—	3	9	no	—
88	C₃	C₃×Q₈×H(3,𝔽₃)	1	1	yes	characteristic
89	C₃	C₃×Q₈×3₋¹⁺²	1	3	yes	—
90	C₂	C₂²×#6891	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃⁴	C₂²×C₃
LCS	G ▷ C₂×C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃×Q₈×3₋¹⁺²	C₃	semidirect
4. Q₈×C₃²⋊C₉	C₃	semidirect
9. #6891	Q₈	direct (N × H)
15. C₃×Q₈×C₉	C₃²	semidirect
26. C₃×3₋¹⁺²	C₃×Q₈	semidirect
27. C₃²⋊C₉	C₃×Q₈	semidirect
33. C₃²×Q₈	3₋¹⁺²	semidirect
55. C₃×C₉	C₃²×Q₈	semidirect
77. C₃²	Q₈×3₋¹⁺²	semidirect
80. Q₈	#6891	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#6891	2
C₃⁴×Q₈	3
C₃×Q₈×3₋¹⁺²	3
Q₈×C₃²⋊C₉	3

Subgroups — Q8×#6891 #14517836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6891 #14517836